Identity

Reciprocal and quotient relations between the ratios

Trigonometry • Trigonometric Identity

Formula

tan θ = \frac{s i n θ}{c o s θ}

Wherever the denominators are non-zero,

cosec θ = \frac{1}{s i n θ}

sec θ = \frac{1}{c o s θ}

cot θ = \frac{1}{t a n θ}

tan θ = \frac{s i n θ}{c o s θ}

and

cot θ = \frac{c o s θ}{s i n θ}

How to use it

1Rewrite everything in $sin θ$ and $cos θ$ when an expression mixes several ratios; the single fraction that results is almost always easier to handle.
2Multiplying an equation through by $sin θ cos θ$ clears $sec θ$ and $cosec θ$ at one stroke.
3 $tan θ cot θ = 1$ and $sin θ cosec θ = 1$ turn many products into $1$ on sight.

Simplify

\frac{c o t θ}{cosec θ}

Step-by-step

1
$\frac{c o t θ}{cosec θ} = \frac{\frac{c o s θ}{s i n θ}}{\frac{1}{s i n θ}}$
Why: Write both ratios in terms of $sin θ$ and $cos θ$ .
2
$= \frac{c o s θ}{s i n θ} \times sin θ = cos θ$

Answer

cos θ

Watch out

Each relation fails where its denominator vanishes:

sec θ

and

tan θ

are undefined at

θ = 9 0^{\circ}

, and

cosec θ

and

cot θ

θ = 0^{\circ}

. Multiplying an equation by

sin θ cos θ

is safe only because writing down

sec θ

cosec θ

already asserts that neither factor is zero.

Also known as

reciprocal trigonometric ratios, quotient identity, sec cosec cot in terms of sin and cos, tan equals sin over cos